#7718257

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

(4,8)(6,7)(9,10,13,11)(12,15,16,14)(18,20)(22,23);(1,2)(3,5)(4,6)(7,8)(17,18,21,20)(19,22,24,23);(2,3)(6,7)(10,11)(14,15)(17,18,21,20)(19,23,24,22);(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,22,20,23);(1,3,5,2)(4,7,8,6)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22);(1,4)(2,6)(3,7)(5,8)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7625453);